Question

The coefficient of the term independent of x in the expansion of ${(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$

• A. 70
• B. 112
• C. 105
• D. 210

Answer 1

The correct option is D 210

Given expression$=\frac{(x^{1/3}{)}^3+(1{)}^3}{(x^{2/3}-x^{1/3}+1)}-\frac{(x-1)}{x^{1/2}(x^{1/2}-1)}$
$=\frac{(x^{1/3}+1)+\left(x^{2/3-x^{1/3}+1}\right)}{x^{2/3}-x^{1/3}+1}-\frac{(x^{1/2}+1)(x^{1/2}-1)}{x^{1/2}(x^{1/2}-1)}$
$=(x^{1/3}+1)-(1+x^{-1/2})=x^{1/3}-x^{-1/2}$
$\Rightarrow {(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$
$={(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$
$=(x^{1/3}-x^{1/2}{)}^{10}$
$T_{r+1}in(x^{1/3}-x^{1/2}{)}^{10}is$
${}^{10}{C}_r(x^{1/3}{)}^{10-r}.(-1{)}^r.(x^{-1/2}{)}^r$
$=(-1){{}^r}^{10}{C}_r{x}^{(\frac{10-r}{3}-\frac{r}{2})}$which is independent of x
If $(\frac{10-r}{3}-\frac{r}{2})=0\Rightarrow r=4$
Hence required coefficient ${=}^{10}{C}_4(-1{)}^4=210$